On Sep 1, 2006, at 9:01 AM, Ruby Quiz wrote:

> If a number is not happy than it is obviously unhappy. Now that you  
> have this
> program, what is the largest happy number you can find?

Clearly, there is no largest happy number. Any number of the form  
10**n in base b is b-relative happy (with rank 1). I feel this makes  
the stated question an ill-conditioned one.

> What is the happiest number between 1 and 1,000,000. I define the  
> happiest number
> as the smallest number that finds the most other happy numbers with  
> it, i.e. 7
> found four other numbers (49, 97, 130, and 10) making it a rank 4  
> in happiness.

This seems a better question to pursue than the first. Other  
questions one might explore: how does rank vary as the numbers  
increase? Does maximum rank grow as happy numbers get bigger? Is  
there a pattern? Are there interesting statistics?

Regards, Morton